A mass conservative generalized multiscale finite element method applied to two-phase flow in heterogeneous porous media

摘要

In this paper, we propose a method for the construction of locallyconservative flux fields through a variation of the Generalized MultiscaleFinite Element Method (GMsFEM). The flux values are obtained through the use ofa Ritz formulation in which we augment the resulting linear system of thecontinuous Galerkin (CG) formulation in the higher-order GMsFEM approximationspace. In particular, we impose the finite volume-based restrictions throughincorporating a scalar Lagrange multiplier for each mass conservationconstraint. This approach can be equivalently viewed as a constraintminimization problem where we minimize the energy functional of the equationrestricted to the subspace of functions that satisfy the desired conservationproperties. To test the performance of the method we consider equations withheterogeneous permeability coefficients that have high-variation anddiscontinuities, and couple the resulting fluxes to a two-phase flow model. Theincrease in accuracy associated with the computation of the GMsFEM pressuresolutions is inherited by the flux fields and saturation solutions, and isclosely correlated to the size of the reduced-order systems. In particular, theaddition of more basis functions to the enriched multiscale space producessolutions that more accurately capture the behavior of the fine scale model. Avariety of numerical examples are offered to validate the performance of themethod.